AN UPPER BOUND FOR THE p NORM OF A GCD-RELATED MATRIX

We find an upper bound for the p norm of the n × n matrix whose i j entry is (i, j)s/[i, j]r, where (i, j) and [i, j] are the greatest common divisor and the least com-mon multiple of i and j and where r and s are real numbers. In fact, we show that if r> 1/p and s < r − 1/p, then ‖((i, j)s/[i, j]r)n×n‖p < ζ(r p)2/pζ(r p − sp)1/p/ζ(2r p)1/p for all positive integers n, where ζ is the Riemann zeta function. Copyright © 2006 Pentti Haukkanen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1